{
  "id": "2209.04740",
  "version": "v1",
  "published": "2022-09-10T20:30:00.000Z",
  "updated": "2022-09-10T20:30:00.000Z",
  "title": "Inducibility in the hypercube",
  "authors": [
    "John Goldwasser",
    "Ryan Hansen"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $Q_d$ be the hypercube of dimension $d$ and let $H$ and $K$ be subsets of the vertex set $V(Q_d)$, called configurations in $Q_d$. We say that $K$ is an \\emph{exact copy} of $H$ if there is an automorphism of $Q_d$ which sends $H$ onto $K$. Let $n\\geq d$ be an integer, let $H$ be a configuration in $Q_d$ and let $S$ be a configuration in $Q_n$. We let $\\lambda(H,d,n)$ be the maximum, over all configurations $S$ in $Q_n$, of the fraction of sub-$d$-cubes $R$ of $Q_n$ in which $S\\cap R$ is an exact copy of $H$, and we define the $d$-cube density $\\lambda(H,d)$ of $H$ to be the limit as $n$ goes to infinity of $\\lambda(H,d,n)$. We determine $\\lambda(H,d)$ for several configurations in $Q_3$ and $Q_4$ as well as for an infinite family of configurations. There are strong connections with the inducibility of graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-10T20:30:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "configuration",
      "inducibility",
      "vertex set",
      "exact copy",
      "cube density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}